Simulation and experiments for optimizing the sensitivity of curved D -type optical fiber sensor with a wide dynamic range

Optics Communications 341 (2015) 210–217

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Simulation and experiments for optimizing the sensitivity of curved Dtype optical fiber sensor with a wide dynamic range Jin-Cherng Hsu a,b, Shau-Wei Jeng a, Yung-Shin Sun a,n a b

Department of Physics, Fu-Jen Catholic University, New Taipei City 24205, Taiwan R.O.C Graduate Institute of Applied Science and Engineering, Fu-Jen Catholic University, New Taipei City 24205, Taiwan R.O.C

art ic l e i nf o

a b s t r a c t

Article history: Received 20 August 2014 Received in revised form 9 December 2014 Accepted 12 December 2014 Available online 16 December 2014

A curved D-type optical fiber sensor (OFS) based on surface plasmon resonance (SPR) is proposed. The parameters, such as the curvature of the fiber, the unpolished depth of the fiber core, the thickness of the deposited gold films, and the incident angle of light upon the gold-deposited surface, affect the sensitivity of this curved OFS. Calculation, simulation, and experiments are performed to optimize the sensitivity by choosing suitable parameters. The feasibility of this curved D-type OFS is demonstrated by monitoring the SPR phenomena of ethylene glycol solutions with different refractive indices. This OFS is applicable as a biosensor for label-free, real-time, and in-situ characterization of biomolecular interactions. & 2014 Elsevier B.V. All rights reserved.

Keywords: Optical fiber sensor Surface plasmon resonance Biosensor Kretchmann's configuration

1. Introduction Surface plasmon resonance (SPR) refers to the consequence of exciting a surface-bound electromagnetic wave at the interface between a metal and transparent material. This technique has been used widely in biosensors for label-free and real-time detection of biomolecular interactions [1–6]. Most of the SPR sensors employ the Kretchmann's configuration where a glass prism is used to achieve total internal reflection (TIR) [7]. Such configuration is applied in both SPR spectroscopy (for real-time measurements) and SPR microscopy (for real-time measurements and endpoint imaging) [8–12]. As an alternative to the prism-based Kretchmann's configuration, the metal surface is modulated with a periodic grating to achieve grating-coupled SPR. The light diffracted by the grating has different orders with at least one inducing SPR [13–15]. In above-mentioned Kretchmann's and grating configurations, bulk optical components and/or complicated fabrication processes are required, which prohibit their in vivo applications. Fortunately, optical fibers, having unique characteristics such as small size, light weight, low cost, flexibility, and easy manipulation, provide a TIR condition for SPR platforms. Various types of optical fiber sensors (OFSs) based on SPR have been proposed, fabricated, and n Correspondence to: Department of Physics, Fu-Jen Catholic University. No. 510, Zhongzheng Rd., Xinzhuang Dist., New Taipei City 24205, Taiwan R.O.C. Tel.: þ 886 2 29052585; fax: þ886 2 29021038. E-mail address: [email protected] (Y.-S. Sun).

http://dx.doi.org/10.1016/j.optcom.2014.12.038 0030-4018/& 2014 Elsevier B.V. All rights reserved.

applied in refractive index, pressure, and temperature measurements [16,17]. Some have also been used as biosensors for detecting biomaterials and biomolecules [18–20]. In 1990, Villuendas and Pelayo reported one of the first SPR-based OFSs [21]. The excitation and detection conditions were analyzed in different wavelength transmission windows, and the geometrical configuration optimizing the sensor head-fiber coupling was proposed. Soon after, Jorgenson and Yee fabricated a fiber-optic chemical sensor by removing a section of the fiber cladding and symmetrically depositing a thin layer of highly reflecting metal onto the fiber core [22]. The sensitivity and dynamic range of this OFS were obtained from experimental measurements of the refractive indices of aqueous solutions, and the results were in good correspondence with those from theoretical models. Afterward, a number of studies have been reported to incorporate SPR into optical fibers for being sensors. Examples include self-assembled monolayers (SAMs)-deposited OFSs [23], metal-coated tapered OFSs [24], single-mode polarization maintaining fiber (PMF)-based OFSs [25], nanoparticles-incorporated OFSs [26,27], fiber Bragg gratings (FBGs)-based OFSs [28,29], and photonic crystal waveguide-based OFSs [30]. D-shaped OFSs with cladding partially polished and D-type OFSs with core side-polished have also been presented for their easy fabrication and high sensitivity. Lo et al. proposed a highly sensitive polarimetric strain sensor based on a four-layered D-shaped optical fiber and SPR technology [31]. The phase difference between the p- and s-polarized waves after SPR coupling was measured using a common-path heterodyne interferometer. The corresponding change in refractive index was

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determined and the strain was then derived. A fiber optical liquid refractometer based on D-type OFS and TIR heterodyne interferometry was reported by Chiu et al. [32] The same group also modified the system by depositing gold thin films on the flat side of the fiber to achieve SPR coupling [33–35]. In 1996, the first curved D-type OFS, reported by Homola and Slavik, was claimed to have a sensitivity of 2  10  5 RIU in the refractive index range of 1.41–1.42 [36]. Such devices are considered to be more sensitive, more accurate, and more stable than other types of OFSs [17]. However, the simulation and fabrication of these curved OFSs are much more complicated than others due to the nature of curvature. The numbers of reflections in the curved core are different with various curvatures and, the incident angles after reflections on the flat-polished surface in a curve are also different. In this paper, we controlled the curvature and the depth of polishing of the curved D-type fiber and evaluated the number of reflections within the fiber sensing region. The reflectance at combined parameters of the gold-film thickness, incident angle, and refractive index of ambient are simulated using an optical thin-film software. Simulation was also performed to acquire the dependence of the number of reflections on the curvature of the fiber and the depth of polishing. Moreover, the fabrication processes of this OFS, including polishing and gold film sputtering, are described. Experimentally, this sensor was applied to monitor the SPR phenomena of ethylene glycol solutions with different refractive indices. The optimal sensitivities derived from simulation and measurements are discussed.

2. Theoretical: principle of surface plasmon resonance Fig. 1(a) shows the schematic drawing of a D-type optical fiber, where the core of the fiber is side-polished and then coated with a thin gold film. The structure of the fiber can be considered as the three-layered Kretchmann's configuration, where an evanescent wave propagates along the metal-dielectric interface as shown in Fig. 1(b). At a certain incident angle θinc, this evanescent wave can interact with the plasma waves on the surface, excite the plasmons, and hence cause resonance. Under such condition, θinc is also called the SPR angle. From Maxwell's equations, the coefficients of reflection for p- and s-polarizations can be expressed as [34,35,37–39]. h r123 =

h h i2k z 2 d2 r12 + r23 e h h i2k z 2 d2 1 + r12 r23 e

where rijh =

Eih − E hj Eih + E hj

,

(1)

, d2 is the thickness of the metal layer, and h¼ p or

211

s. Eipororj s can be further expressed as

E wh

⎧ n2 ⎪ w , h = p, = ⎨ k zw ⎪ ⎩ k zw, h = s ,

w = i or j;

i, j = 1, 2, or 3, (2)

where n1, n2, and n3 are the refractive indices of the glass prism (the core of the fiber), the metal layer (the gold film), and the medium, respectively. For the simulation, we used optical thinfilm software to simulate the SPR. The Snell's law also holds in the metal Au film, which has a large complex value of optical index in visual light. In Eq. (1), the coefficient of reflection for p-polarized light component can be expressed as

r12 + r23 eiδ

rp =

1 + r12 r23 eiδ

,

(3)

where

r12 =

n2 cos (θ1) − n1 cos (θ2 ) , n2 cos (θ1) + n1 cos (θ2 )

(4)

r23 =

n3 cos (θ2 ) − n2 cos (θ 3 ) , n3 cos (θ2 ) + n2 cos (θ 3 )

(5)

and the phase angle

δ=

4π d n22 − n12 (sin θ1)2 . λ

(6)

In Eqs. (4)–(6), θ1 is the incident angle, θ2 ¼ sin  1[(n1/n2)sin θ1], and θ3 ¼sin  1[(n1/n3)sin θ1]. By inputting λ, d, n1, n2, n3, and θ1 parameters into the optical thin-film software, the reflectance R can be derived as

⎛ r + r eiδ 12 23 R = rp2 = ⎜⎜ ⎝ 1 + r12 r23 eiδ

⎞2 ⎟ ⎟ ⎠

(7)

Eq. (7) is the master equation for the reflection simulation of the OFS in the sensing region.

3. Experimental processes 3.1. Fabrication First, we created some trenches of outward curvature whose radius was 480 mm on 4 cm  4 cm  0.5 cm glass slides. Then the fibers were pushed into the bottom of the trenches in 275–375 μm depth against the surface of the glass slide. We used optical

Fig. 1. (a) The scheme of D-type optical fiber sensor. (b) The Kretchmann's configuration.

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Fig. 4. The experimental setup of the curved D-type optical fiber.

3.2. Measurement setup of OFS

Fig. 2. The scheme for the calculation of the number of reflections and the depth of polishing in a curved D-type optical fiber.

matching oil to fill the trenches, and then cured the oil with UV light before the polishing process as shown in Fig. 2. 3.1.1. Side-polishing the fiber core We used multi-mode optical fibers with a core diameter of 62.5 μm, a cladding diameter of 125 μm, and a numerical aperture (NA) of 0.272. A planet-type grinding machine was used to grind and polish the fiber-embedded glass substrates as shown in Fig. 3 (a). When we coupled a He–Ne laser light to the partially corepolished fiber, a leak of light occurred in the desired sensing region as shown in Fig. 3(b). The length of the light leak (2L) as shown in Fig. 2 was the distance between the two positions that the core images, as shown in Fig. 3(c), disappear in the cladding of the polished fiber. The depth of polishing (Y) could be determined by

Y=R−

R2 − L2 ,

Fig. 4 shows the measurement setup containing a light source, a curved D-type OFS, a spectrometer, and a photomultiplier. A tungsten halogen lamp (LS-1 from Ocean Optics, USA) was used as an unpolarized light source whose spectrum ranges between 400 and 900 nm. This light source was directly coupled to the OFS and reflected in the polished region of the fiber core at the incident angle ranged about from 80° to 90°. The spectrometer (Triax 320 from HORIBA, Japan) measured the light coming out from the OFS. The scanning wavelength was set to vary from 400 to 900 nm with a resolution of 0.06 nm. The output light from the spectrometer was sent to a PMT (R5108 from Hamamatsu Photonics, Japan) having a detection range of 400–1200 nm and a strongest response at 800 nm. The electronic signals in an integration time of 0.1 s were then amplified and processed by a computer. The sensing region at the top of the OFS was sequentially added by six different refractive-index liquids, from n1 to n6, which were made by mixing various amount of ethylene glycol with water [40,41]. The refractive indices of these solutions were measured, by an Abbe refractometer, to be 1.3325, 1.3510, 1.3705, 1.3898, 1.4099, and 1.4239. Then, the spectra from all samples were collected for comparison.

(8)

where R is the radius of the curved fiber. 3.1.2. Gold film sputtering Before sputtering, the chamber was pumped to a background pressure of 2  10  6 Torr. Ar gas was fed into the chamber at a rate of about 16 sccm by a mass flowmeter to keep the working pressure at about 2.0  10  3 Torr. Then various thicknesses of the gold films were deposited on the fiber-embedded glass substrates by DC sputtering with a 60-W power and in-situ monitored with a quartz crystal thickness monitor. All thicknesses were ex-site checked again with a VASE M-2000U ellipsometer made by J. A. Woollam Company.

4. Results of simulation 4.1. Simulation of various thicknesses of gold films The SPR phenomenon was analyzed, with gold thin films deposited on the surface of the D-type fiber having a core refractive index of 1.51, by an optical thin-film software (the Essential Macleod software package from Thin Film Center Inc., USA). Fig. 5(a)– (f) plots the reflectance against the incident angle under different ambient indices at different thicknesses of gold films (d) from 5 to 30 nm. For simplifying the simulation process, a p-polarized light with a wavelength of 700 nm was used in this process. This wavelength was also used in finding the optimal sensitivity of the

Fig. 3. (a) The picture of an embedded optical fiber. (b) The light leak from coupling a 632 nm laser light into the fiber. (c) The picture shows the core images disappearing in the cladding of the polished fiber.

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213

Fig. 5. The reflectance against the incident angle under different ambient indices at d ¼ (a) 5 nm, (b) 10 nm, (c) 15 nm, (d) 20 nm, (e) 25 nm, and (f) 30 nm.

OFSs experimentally. For d ¼5–15 nm, local minima occurred at incident angles between 70° and 85°, and this resonant angle increased with increasing refractive index. Moreover, the locally minimum reflectance decreased with increasing refractive index. However, for d¼ 20–30 nm, the resonant angle as well as the minimum reflectance increased with increasing refractive index. Therefore, there may be an optimal thickness between 10 nm and 15 nm due to the reverse results observed in the above modeling processes. 4.2. Calculation of the number of reflections Based on the calculation in supplementary, the number of reflections and the incident angles after each reflection were found by giving initial parameters. Table 1 shows the simulation results of an optical fiber with the first incident angle (X)¼89°, the radius of curvature of the fiber (R) ¼480 mm, the remaining unpolished depth of the fiber core (t)¼ 30 μm, and the half-length of the sensing region (L)¼9550 μm. A negative L5 occurs as the 5th reflection light passes through the symmetry line KO in Fig. 2. Therefore, the total number of reflections is 2  4þ 1 ¼9 due to 4 times in two symmetry positions of the sensing region and one in the middle position of the 5th reflection. Simulations on optical fibers with different R (480, 530, 580, and 630 mm) and different t

(10, 20, 30, 40, and 50 μm) were also performed (data not shown). 4.3. Reflection simulation of a sample The final reflectance through a number of reflections was found by returning the incident-angle values derived in Section 4.2 back to the software (Section 4.1). The initial intensity was set to be 100%, and the value decreased successively after each reflection. The incident angles are 89, 87.6, 86.8, 86.4, 86.2, 86.4, 86.8, 87.6, and 89 sequentially as shown in Table 1. The reflectance (R%) at these angles was simulated by the thin film software. Then the right-bottom part of this table shows the final reflectance after 9-times reflections. Simulations on optical fibers with different R (480, 530, 580, and 630 mm), different t (10, 20, 30, 40, and 50 μm), different d (5, 10, 15, 20, 25, and 30 nm), and different n (1–1.4239) were also performed, which include 840 different conditions in total (data not shown). 4.4. Simulation of sensitivity The average attenuated intensity ( Ani ) at the reflection index of ni is defined as 100% – (average final reflectance). For example, in Table 1, the average final reflectance is 59.73%, and the average attenuated intensity is 40.27%. If t is larger than 30 μm, the

Table 1 The calculated final reflectance of an optical fiber with R¼ 480 mm, t¼ 30 μm, L1 ¼ 9550 μm, and d ¼ 5 nm. n6 ¼1.4239, and optical wavelength¼700 nm. n

Xn (deg)

Ln (μm)

θn (deg)

Yn (deg)

Xn þ 1 (deg)

Ln þ 1 (μm)

Intensityn (%)

R%n (%)

Final R%n (%)

1 2 3 4 5 6 7 8 9

89.00 87.61 86.83 86.38 86.16 86.38 86.83 87.61 89.00

9550 4250 2522 1377 431 – – – –

1.14 0.51 0.301 0.161 0.051 – – – –

88.3 87.2 86.6 86.3 86.2 – – – –

87.6 86.8 86.4 86.2 86.2 – – – –

4250 2522 1377 431  464 – – – –

100 96.32 86.15 74.33 64.14 55.34 47.75 41.20 36.85

96.32 89.44 86.28 86.28 86.28 86.28 86.28 89.44 96.32

96.32 86.15 74.33 64.14 55.34 47.75 41.20 36.85 35.49

Final reflectance: Final R%n ¼Intensityn  R%n. Average final reflectance (Final R% ) ¼59.73%. A1.4239 ¼ 100  59.73% ¼ 40.27%.

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Table 2 The simulated rms values of ΔA/Δn for the selected optical fibers.

S=

R (mm)

t (μm)

d (nm)

ΔA/Δn

Sensitivity (RIU)

480 480 480 480 480 530 530 530 530 530 580 580 580 580 580 630 630 630 630

10 10 20 20 30 10 10 20 20 30 10 10 20 20 30 10 20 10 20

10 15 10 15 10 10 15 10 15 10 10 15 10 15 10 10 15 10 15

3.30 4.92 3.95 4.01 3.27 3.95 4.01 3.95 4.01 3.27 3.95 4.01 3.88 3.29 3.27 3.87 4.37 3.89 3.29

3.03  10  5 2.03  10  5 2.53  10  5 2.49  10  5 3.06  10  5 2.53  10  5 2.49  10  5 2.53  10  5 2.49  10  5 3.06  10  5 2.53  10  5 2.49  10  5 2.58  10  5 3.04  10  5 3.06  10  5 2.59  10  5 2.29  10  5 2.57  10  5 3.04  10  5

⎡ A ni + 1 − An i ⎤2 ⎥ , n − ni ⎥⎦ i=0 ⎣ i+1 6

∑ ⎢⎢

(10)

where ΔT is the resolution of the intensity of the light source and N is a normalization factor. The intensity resolution of our measurement system was experimentally determined to be 0.01%, and N is unity here. The sensitivity is optimal in fiber sensors with t ranging from 10 to 30 μm and d from 10 to 15 nm. The best sensitivity of 2.03  10  5 RIU (ΔA/Δn ¼4.92) is found in an OFS with R¼480 mm, t ¼10 μm, and d ¼15 nm.

5. Experimental results

attenuated intensity is higher than 80%. Regardless of other parameters, as the remaining unpolished depth (t) of the fiber increased, the number of reflections as well as the attenuated intensity increased. Although the average attenuated intensity could reach 95% for t¼ 50 μm, the SPR signal is not the most sensitive. Therefore, the attenuation slope for a specific set of parameters R, t, and d, is defined as the root-mean-square (rms) value,

ΔA/Δn =

ΔT , N (ΔA/Δn)

(9)

where n0 ¼1, n1 ¼1.3325, n2 ¼1.3510, n3 ¼ 1.3705, n4 ¼ 1.3898, n5 ¼1.4099, and n6 ¼ 1.4239. In Eq. (9), the average attenuated intensity (A) is normalized to unity (100%-1). Table 2 lists several sets with different parameters that provide ΔA/Δn values higher than 3.0. It is clear that the higher the slope ΔA/Δn is, the more obvious the SPR phenomenon is. Furthermore, we can evaluate the sensitivity of the OFS as [34]

Thin gold films of various thicknesses (d)¼ 29.3, 23.8, 16.6, and 12.0 nm were deposited on the flat-polished surfaces of the curved D-type optical fibers with radius (R)¼480 mm and core-unpolished depth (t) ¼30.7 μm. The SPR phenomena were observed as adding different refractive-index liquids (n) on top of the sensing region (data not shown). There is no significant relation between the refractive index and the intensity of transmitted light at d¼ 29.3 and 23.8 nm. From the spectra of d ¼16.6 nm and 12.0 nm, the intensity decreased with increasing refractive index except for n¼ 1.4239 in the case of d ¼16.6 nm. It is consistent with the SPR theory and, therefore, d ¼12 nm was concluded to be an optimal thickness of gold films. Next, to find the optimal sensitivity, optical fibers with R¼480 mm, d ¼12.0 nm, and t¼0, 10.0, 20.1, 31.5, 41.1, and 55.4 μm were fabricated and tested. Fig. 6(a)–(f) show the SPR spectra of these fibers. In Fig. 6(a), no clear relationship between the refractive index and the intensity of transmitted light was simply observed because SPR did not occur at unpolished (t ¼0) fibers. For t ¼10.0–55.4 μm, the intensity decreased with increasing refractive index. Again, using Eq. (10), the sensitivity S (in RIU) of the OFS can be experimentally derived. Referring to the spectrum of the light source, the maximum intensity should be normalized to unity in the absence of SPR, so a normalization factor N of 2.9 was used. To find ΔA/Δn at different t values, instead of looking for the peak value in each curve, a particular wavelength was selected for plotting the intensity against the index. For example, in each panel of Fig. 6, the intensities at wavelength¼700 nm were picked and plotted against the refractive index, as shown in Fig. 7. The slope was negative because

Fig. 6. The spectra of optical fibers with R¼ 480 mm, d ¼12.0 nm, and t¼ (a) 0 μm, (b) 10.0 μm, (c) 20.01 μm, (d) 31.5 μm, (e) 41.1 μm, and (f) 55.4 μm.

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215

Table 3 A comparison of the current OFS and other similar ones in experimental sensitivity and dynamic range.

Fig. 7. The intensities and slopes of the curved D-type optical fiber at wavelength¼ 700 nm versus the refractive index under different t values.

Fig. 8. The slope (ΔA/Δn) at all wavelengths under different t values.

the intensity decreased with increasing index. The OFS at t¼ 31.5 μm has a maximum slope of  0.168, which corresponded to a sensitivity of  2.05  10  4 RIU after normalization. Similarly, ΔA/Δn at other wavelengths was calculated for OFSs with different t values as shown in Fig. 8. For all curves, peak values (maximum slopes) occurred at wavelength 700 nm, and for R¼ 480 mm and d¼ 12.0 nm fibers, the sensitivity was best at t¼ 31.5 μm.

6. Discussion In this study, multi-mode optical fibers were used because they provide a higher “light-gathering” capacity and easier processing to controlled parameters comparing with single-mode fibers. Our curvature-trench method has the following advantages: (1) the glass substrate used to fix the optical fiber can prevent the core from breaking during the polishing; (2) the curvature of the fiber can be precisely controlled for calculating the depth of polishing. The current curved D-type sensor has a wide dynamic range of 1.33–1.43 refractive indices with sensitivities of 2.03  10  5 RIU theoretically and 2.05  10  4 RIU experimentally. Such difference (about one order of magnitude) was also found in theoretical and experimental results of D-type optical fiber sensors [35,42]. These

Type of OFS

Sensitivity in RIU

Dynamic range in refractive indices

Reference

Curved D-type OFS Curved D-type OFS D-type OFS D-type OFS D-type OFS D-shaped OFS D-shaped OFS U-shaped OFS

2  10  4

1.33–1.43

This work

5

1.41–1.42

[36]

1.33–1.37 Near 1.40 Near 1.37 1.33–1.40 1.33–1.40 1.33–1.35

[33] [34] [42] [43] [44] [19]

2  10

2  10  4 8  10  5 3  10  5 1  10  3–3  10  4 3  10  4 4  10  4

two values of theoretical predictions and experimental results are different mainly because in simulation the incident angle was set to 89° for simplicity but in experiments this angle could range from 80° to a value slightly less than 90°. Another possibility is that a p-polarized light was used in simulations but experimentally an unpolarized light was applied. In the latter case, light components other than the p-polarized one could present high background signals and therefore lower the sensitivity. However, comparing to other OFSs using p-polarized laser light sources, it is much easier to set up and operate the current OFS-spectrometer system. With this setup, we can also investigate the SPR phenomenon under different wavelengths. Moreover, from simulation shown in Table 2, for fibers with R¼480 mm and d ¼10 nm, the sensitivity was optimal at t¼20 μm. But experimentally, fibers with t  30 μm yielded a better sensitivity than others. One possible reason is that the fiber with t  30 μm provides a larger crosssection area than that with t 20 μm to enhance the SPR phenomenon. A comparison of the current OFS and other similar ones in experimental sensitivity and dynamic range are shown in Table 3 [43,44]. This table clearly indicates a trade-off between high sensitivity and wide dynamic range. For applications as a biosensor, a wide dynamic range is desirable because the refractive index can change significantly from one biomolecule to another (e.g. from small DNA molecules to whole cells). Our theoretical and experimental results suggest that this curved SPR-based D-type optical fiber can serve as a biosensor for sensitive, label-free, real-time, and in-situ characterization of biomolecular interactions.

7. Conclusion In this paper, a curved D-type optical fiber sensor based on surface plasmon resonance was reported. Detailed calculation was performed to find the number of reflections within the sensing region. Various sets of parameters, including the radius of curvature of the fiber (R), the remaining unpolished depth of the fiber core (t), and the thickness of the deposited gold thin films (d), were used to simulate the best sensitivity of this fiber sensor. The fabrication of such curved optical fibers was then elaborated. For a demonstration, they were applied to monitor the SPR phenomena of ethylene glycol solutions with different refractive indices. Sensitivities of 2.03  10  5 RIU and 2.05  10  4 RIU were derived theoretically and experimentally, respectively. And the dynamic range was from n ¼1.33 to n ¼1.43, which is considered very wide. By virtue of the advantages of this curved D-type OFS, e.g. small size, inexpensiveness, sensitivity, and in vivo measurement, it has great applications as physical, chemical, and biological sensors.

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Acknowledgment

Eq. (S5) can be expressed in terms of angles as

The authors thank financial supports from Taiwan MOST 1022221-E-030-011 (J. C. Hsu) and MOST 101-2112-M-030-003-MY3 (Y. S. Sun).

The number of reflections in the sensing region affects the intensity of transmitted signals from the output port of OFS. The more the number of reflections, the higher the signal is attenuated due to SPR [34,35]. However, in a curved D-type OFS, the incident angle changes after each reflection and depends on the curvature, the length of the sensing region, the thickness of the remaining core, and the first incident angle upon arriving the sensing region. The number of reflections was evaluated according to the below parameters defined in Fig. 2. Point O: the center of curvature; Point A: the beginning of the sensing region (the first incident point on the sensing region); Point D: the first incident point on the curved surface; Point B: the second incident point on the sensing region; Point C: node of the vertical line AC and horizontal line OC ; Point E: node of the vertical line BE and horizontal line OE ; Line KP : remaining unpolished part of the fiber core, length ¼t; Line KO, AO and BO curvature of the fiber, length ¼ R; Line AK : half of the sensing region, length ¼L; Line CO: length ¼L; Line EO : length ¼ L1; Angle ∠CAD: the first incident angle on the sensing region, angle ¼X; Angle ∠ADB: double of the first incident angle on the curved surface, angle ¼ 2Y; Angle ∠DBE : the second incident angle on the sensing region, angle ¼X1; Angle ∠CAO: the angle between lines AO and AC , angle ¼ θ; Angle ∠AOD: the angle between lines AO and DO , angle ¼Y  Xþ θ; Among these parameters, t, R, L, and X can be controlled experimentally, and the goal of the following calculation is to find the incident angle Xn after n times of reflections on the sensing region. In ΔADO,

(S1)

Eq. (S1) can be expressed in terms of lengths as

sin (X − θ) sin (180° − Y ) . = R L2 + (R + t)2

The new X1 can be found to be

X1 = 2Y − X .

sin (X1 − θ1) sin (180° − Y1) . = R L12 + (R + t)2

where in ΔACO,

θ=

tan−1

⎧ ⎫ ⎪ L12 + (R + t)2 ⎪ × sin (X1 − θ1) ⎬ , Y1 = sin−1⎨ ⎪ ⎪ R ⎩ ⎭

(S9)

where

⎡ L1 ⎤ θ1 = tan−1⎢ ⎥, and ⎣R + t ⎦

(S10)

X2 = 2Y1 − X1.

(S11)

In Eq. (S9)–(S11), X1, R, and t are known, but L1 remained to be determined. L1 is related to L as

L1 = L − AB.

(S12)

In ΔABD,

sin (∠ADB) sin (∠ABD) = . AB AD

(S13)

Inserting angles, Eq. (S13) can be express as

sin (2Y ) sin (90° − X1) = , AB AD

(S14)

where

sin (90° − X1) = cos (X1).

(S15)

Using X1 ¼2Y X, Eq. (S15) can be express as

sin (2Y ) cos (2Y − X) = AB AD

(S16)

Again, in ΔADO,

sin (∠AOD) sin (∠DAO) , and = AD DO

(S17)

sin (Y − X + θ) sin (X − θ) = . R AD

(S18)

AD =

R sin (Y − X + θ) . sin (X − θ)

(S19)

Using Eq. (S19), Eq. (S16) can be expressed as

(S3)

AB =

R sin (2Y ) sin (Y − X + θ) . sin (X − θ) cos (2Y − X)

(S20)

And Eq. (S12) gives

(S4)

Y can then be found after inserting L, R, t, and X. In ΔADO,

∠ADB + ∠BAD + ∠ABD = 180°.

(S8)

In Eq. (S18), AD can be expressed as

(S2)

θ can be expressed as

⎡ L ⎤ ⎢⎣ ⎥. R + t⎦

(S7)

Similarly, solving Eq. (S8), Y1 can be expressed as

Solving Eq. (S2), Y can be expressed as

⎧ ⎫ ⎪ ⎪ L2 + (R + t)2 × sin (X − θ) ⎬ , Y = sin−1⎨ ⎪ ⎪ R ⎩ ⎭

(S6)

The same procedure can be used to derive X2, the third incident angle on the sensing region. Eq. (S2) can be expressed as

Appendix. : Calculation of the number of reflections

sin (∠DAO) sin (∠ADO) . = DO AO

2Y + (90° − X) + (90° − X1) = 180°.

(S5)

L1 = L −

R × sin (2Y ) × sin (Y − X + θ) . sin (X − θ) × cos (2Y − X)

(S21)

Using Eq. (S9)–(S11), this L1 can be used to derive X2. Similar to Eq. (S21), Ln can be expressed in terms of Xn  1, θn  1, Yn  1, and Ln  1 as

J.-C. Hsu et al. / Optics Communications 341 (2015) 210–217

L n = L n− 1 −

R × sin (2Yn − 1) sin (Yn − 1 − X n − 1 + θ n − 1) . sin (X n − 1 − θ n − 1) cos (2Yn − 1 − X n − 1)

and a new

(S22)

θn can be solved as

⎡ Ln ⎤ θn = tan−1⎢ ⎥. ⎣R + t ⎦

(S23)

Finally, Yn and Xn can be solved using

⎧ ⎫ ⎪ L n2 + (R + t)2 ⎪ Yn = sin−1⎨ sin (X n − θn ) ⎬ , and ⎪ ⎪ R ⎩ ⎭

(S24)

X n = 2Yn − 1 − X n − 1.

(S25)

As the first incident angle on the sensing region (X), the radius of the curved fiber (R), the length of remaining unpolished the fiber core (t), and the length of the sensing region (2L) are known, the number of reflections and the incident angles after reflections can be derived.

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